Chapter 21

Wigner matrices

G. Ben Arous1, A. Guionnet2

1Courant Institute of the Mathematical Sciences, New York University,

251 Mercer street, NY10012, USA2Unite de Mathematiques Pures et Appliquees, Ecole Normale Superieure,

Lyon 69007, France

Abstract

This is a brief survey of some of the important results in the study of the

eigenvalues and the eigenvectors of Wigner random matrices, i.e. random her-

mitian (or real symmetric) matrices with i.i.d entries. We review briefly the

known universality results, which show how much the behavior of the spectrum

is insensitive to the distribution of the entries.

21.1 Introduction

In the fifties, Wigner introduced a very simple model of random matrices to

approximate generic self-adjoint operators. It is given as follows. Consider

a family of independent, zero mean, real or complex valued random variables

{Zi,j}1≤i<j , independent from a family {Yi}1≤i of i.i.d centered real-valued ran-

dom variables. Consider the (real-symmetric or hermitian) N × N matrix XN

with entries

XN (j, i) = XN (i, j) =

{

Zi,j , if i < j,Yi , if i = j.

(21.1.1)

We call such a matrix a Wigner matrix, and if the random variables Zi,j and

Yi are Gaussian, we use the term Gaussian Wigner matrix. The case of Gaus-

sian Wigner matrices in which EY 21 = 2 and E|Z1,2|2 = 1 is of particular

importance, since their law is invariant under the action of the orthogonal,

1

2 CHAPTER 21. WIGNER MATRICES

(resp. unitary) group if the entries are real (resp. complex), see e.g. Chap-

ter 3 in this handbook. In the Gaussian case, the distribution of the matrix

is invariant by the action of the natural symmetry group SO(N) (respectively

SU(N)), the eigenvalues of the matrix XN are independent of the eigenvec-

tors which are Haar distributed. The joint distribution P(β)N of the eigenvalues

λ1(X) ≤ · · · ≤ λN (X) is given by

P(β)N (dx1, . . . , dxN ) = C

(β)N 1x1≤···≤xN

|∆(x)|βN∏

i=1

e−βx2i /4dx1 · · · dxN , (21.1.2)

where

C(β)N =

(

∫ ∞

−∞· · ·∫ ∞

−∞|∆(x)|β

N∏

i=1

e−βx2i /4dxi

)−1

.

In this formula, β = 1 (resp. 2) if the entries are real (resp. complex). The

distributions of the eigenvalues of these random matrices are usually called

the Gaussian orthogonal ensembles (GOE) or the Gaussian Unitary Ensemble

(GUE) respectively. These and more general invariant ensembles are the main

focus of the other chapters of this handbook. The present chapter is devoted

to the study of general non-invariant Wigner ensembles and of their universal

properties, that is the properties they share with the invariant Gaussian en-

sembles, at the global and at the local levels, in the bulk and at the edge of

the spectrum. This is a very wide domain of research, impossible to summarize

with any depth in the format of this review article. The study of the properties

of the spectrum of Wigner ensembles has a very long and rich history, and a few

recent expository works cover the results and their proof in much greater depth

than can be achieved here ([Bai99], [And09]). Compounding the difficulty is the

fact that the field has seen recently a burst of very important results. Indeed,

the question of the universality in the bulk at the local level for the spectrum of

Wigner matrices, which had long been seen as one of the central open problems

in Random Matrix Theory, has been solved recently by two groups of mathe-

maticians (L.Erdos, J.Ramirez, B.Schlein, S.Peche, H.T. Yau and T.Tao and

V.Vu). These new universality results open a very exciting period for the field

of Random Matrix Theory (see the Bourbaki seminar [Gui10]). We will try to

cover succinctly those new developments, without any claim for completeness

given the rapid pace of recent progress. We will give some of the universality

results as well as some results known for the Gaussian ensembles which have

not yet been proved to be universal, i.e. the open questions left after the recent

wave of progress. We will also give some of the known limits of universality, i.e.

some of the cases where it is known that the results differ from the Gaussian

ensembles. Before proceeding let us comment very briefly and informally about

the methods of approach for general non-invariant Wigner ensembles. In fact

21.1. INTRODUCTION 3

there are very few such possible approaches, essentially three. Let us note first

that the main tool used for invariant ensembles, i.e. the explicit computation of

the distribution of the spectrum, is of course not possible for the non-invariant

ensembles. The first general approach for non-invariant Wigner ensembles is

the moment method, which consists in computing the asymptotic behavior of

the moments of the spectral measure LN , that is the empirical measure of the

renormalized eigenvalues of XN , or equivalently the normalized trace of powers

of the random matrix∫

xkdLN (x) =1

Nk2+1

Tr[XkN ]

The second one, the resolvent method, consists in computing the asymptotic

behavior of the normalized trace of the resolvent, i.e the Stieltjes transform of

the spectral measure∫

1

z − xdLN (x) =

1

NTr[(zId − N− 1

2 XN )−1].

For a long time these two methods were basically the only ones available.

The survey [Bai99] shows how far one can go using these tools. Another ap-

proach to universality is based on explicit formulas and concerns a special case

of non-invariant ensembles, the Gaussian divisible case. This is the case of ran-

dom Wigner matrices, where the distribution of the entries is the convolution

of an arbitrary distribution and of a centered Gaussian one. These matrices

can be written as

XN =√

εGN +√

1 − εVN (21.1.3)

with a matrix GN taken from the GOE or the GUE, independent from a self-

adjoint matrix VN , and some ε ∈ (0, 1). Note that when VN is a Wigner matrix,

so is XN . K.Johansson [Joh01] studied such matrices when XN is taken from

the GUE based on rather explicit formulas for the joint law of the eigenvalues

of XN . Another point of view is based on the fact that the spectrum of such

matrices is described by the Dyson Brownian motion, see e.g. [And09, Theorem

4.3.2], at time t = − log ε which is a weak solution to the system

dλNi (t) =

√2√

βNdWi(t) +

1

N

∑

j:j 6=i

1

λNi (t) − λN

j (t)dt , i = 1, . . . , N , (21.1.4)

with initial condition given by the eigenvalues of VN and Wi, 1 ≤ i ≤ N indepen-

dent Brownian motions. This fourth perspective, based on Dyson’s Brownian

motion, has been considerably strengthened and proved to be very useful for far

more general matrices, in the recent work of the group around Erdos, Schlein

and Yau.

Universality can also be proved by using approximation arguments from the

above models, see e.g. [R06, Tao09a, Tao09b].

4 CHAPTER 21. WIGNER MATRICES

21.2 Global properties

In this section, we describe the global properties of the spectrum of Wigner

matrices. It turns out that, when the entries have a finite second moment,

the eigenvalues of XN are of order√

N with overwhelming probability, and

the global properties of the spectrum of XN shall be described by the spectral

measure

LN :=1

N

N∑

i=1

δλNi

(X)√

N

which is a probability measure on the real line. Note that in particular, for any

a < b

LN ([a, b]) =1

N]{i : λN

i (X) ∈√

N [a, b]}is the proportion of normalized eigenvalues falling in the interval [a, b].

The first result of RMT is of course Wigner’s theorem which says that LN

converges towards the semi-circle law. We will give some of the known results

about the fluctuations around this limit. We will also mention concentration

results, and see that the law of LN concentrates under fairly general hypoth-

esis. One first result which seems out of reach for general Wigner ensembles

concerns the large deviations of the spectral measure. Whereas in the case of

the Gaussian ensembles, one knows a full large deviations principle, it seems

impossible at this time to get such a result for any non-invariant ensemble.

We will recall this large deviations principle as well as the moderate deviations

principle for Gaussian ensembles. A moderate deviation has been given for

some non-invariant Gaussian divisible ensembles.

Recently, it was observed [Erd08] that the convergence of the spectral mea-

sure towards the semi-circle law holds in a very local sense, that is can be

obtained on intervals with width going to zero just more slowly than the typ-

ical spacing between eigenvalues. This result thus interpolates between global

and local properties of the spectrum. Moreover, this local convergence is inti-

mately related with the fact that the eigenvectors are delocalized in the sense

that all of their entries are at most of order (log N)4/√

N with overwhelming

probability.

Finally, in section 21.2.5, we will describe the global behavior of the spec-

trum when the entries do no have a finite second moment but have heavy tails,

provided they belong to the domain of attraction of an α-stable law.

21.2.1 Convergence to the semi-circle law

The main result of this section goes back to Wigner [Wig55] and gives the weak

convergence of LN towards the semi-circle law σ given by

dσ(x) =1

2π1[−2,2]

√

4 − x2dx .

21.2. GLOBAL PROPERTIES 5

Theorem 21.2.1 Assume E[|Z1,2|2] = 1 and E[Y 21 ] < ∞. Then, for any

continuous function f with polynomial growth,

limN→∞

∫

f(x)dLN (x) =

∫

f(x)dσ(x) a.s.

This theorem was originally proved for polynomial functions and the conver-

gence in probability, under the condition that the entries have all their moments

finite (see [Wig55] or [And09, Theorem 2.1.1]). However, almost sure conver-

gence can be obtained by using Borel-Cantelli Lemma and fine estimates of the

covariances [And09, Exercise 2.1.16]. Finally, the assumptions that all moments

are finite can be removed by approximation using Hoffman-Wielandt inequality

[And09, Lemma 2.1.19].

The next question is about speed of convergence of the spectral measure

to the semi-circle law. It is not universal, and should depend on the law of

the entries and the metric used to measure it. In [Bai93, Theorem 4.1], it was

shown that

supx∈R

|E[LN ((−∞, x])] − σ([−2, x ∨ −2])| = O(N− 14 )

under the assumption that the entries have a finite fourth moment. When

the sixth moment is finite, it was shown [Bai09, Lemma 6.1] that the speed is

at most of order O(N− 12 ). For complex Gaussian divisible ensembles, it was

proved to be at most of order O(N−ν) for all ν < 2/3 in [Got07, Theorem 1.2].

The conjecture is that the optimal speed is O(N−1), as proved for the GUE

case in [Got05]. A similar question may be asked for the expectation of the

distance between LN and σ. We refer to [Cha04] and references therein.

21.2.2 Fluctuations around the semi-circle law

The standard fluctuations results of the spectral measure for invariant Wigner

ensembles states that for any smooth enough function f the law of

δN (f) = N(

∫

f(x)dLN (x) −∫

f(x)dσ(x))

converges towards a Gaussian variable (cf [Joh98] for the Gaussian ensembles).

Note here the intriguing speed 1/N , to be compared with the classical 1/√

N

speed of the classical central limit theorem. Such a result for non-invariant

ensembles was first proved in the slightly different Wishart matrix model case

by Jonsson [Jon82] for polynomial functions f . We refer the reader to [And09,

Theorem 2.1.31] for similar techniques for Wigner matrices under the hypothesis

that the entries have all their moments finite and the centering is done with

respect to the mean instead of the limit. This last result was extended in

6 CHAPTER 21. WIGNER MATRICES

[Lyt09] under only the finite fourth moment condition. An interesting point

is that the covariance of the limiting Gaussian variable depends on the fourth

moment of the distribution of the entries. The fluctuations result for δN (f) was

generalized, under the assumption of a finite sixth moment, in [Bai09, Theorem

1.1] for functions f with four continuous derivatives. An interesting feature of

[Bai09] is a generalization of the result of [Joh98] for the GOE which shows

that the limiting Gaussian variable is not centered in general in the real case.

Such central limit theorems hold for many models of random matrices, see

e.g. unitary matrices following the Haar measure [Dia94].

21.2.3 Deviations and Concentration properties

Under stronger conditions about the distribution of the entries, more can be said

about the convergence of the spectral measure. One can prove concentration

results. Indeed, it turns out that the evaluation of the spectral measure along

a smooth function is a smooth function of the entries. Therefore, the theory

of concentration of measure applies to such random variables. For instance,

we have the following concentration of measure property. To begin with, recall

that a probability measure P on R is said to satisfy the logarithmic Sobolev

inequality (LSI) with constant c if, for any differentiable function f ,

∫

f2 logf2

∫

f2dPdP ≤ 2c

∫

|f ′|2dP.

When the distribution of the entries satisfy log-Sobolev inequality, not only

the spectral measure concentrates but also the eigenvalues themselves. More

precisely we have the following statement.

Theorem 21.2.2 Suppose that the laws of the independent entries

{XN (i, j)}1≤i≤j≤N all satisfy the (LSI) with constant c/N . Then, for any Lip-

schitz function f on R, for any δ > 0,

P

(

|∫

f(x)dLN (x) − E[

∫

f(x)dLN (x)]| ≥ δ

)

≤ 2e− 1

4c|f |2LN2δ2

. (21.2.1)

Further, for any k ∈ {1, . . . , N},

P(

|f(N− 12 λk(XN )) − Ef(N− 1

2 λk(XN ))| ≥ δ)

≤ 2e− 1

4c|f |2LNδ2

. (21.2.2)

These results can also be generalized to the case when the distribution

of the entries satisfy a Poincare inequality or are simply bounded (but then

the test function f has to be convex). We refer to [Gui00] or [And09] for

precise statements and generalizations. Interestingly, concentration inequalities

hold under the mere assumption of independence (in fact only of the vectors

((XN (ij))j≤i)1≤j≤N ), but with a worst speed estimate [Bord10].

21.2. GLOBAL PROPERTIES 7

Theorem 21.2.3 Assume f has finite variation norm

‖f‖TV := supk∈N

x0<x1<x2···<xk

k∑

`=1

|f(x`) − f(x`−1)|.

Then, for any δ > 0,

P

(

|∫

f(x)dLN (x) − E[

∫

f(x)dLN (x)]| ≥ δ‖f‖TV

)

≤ 2e−12Nδ2

. (21.2.3)

The proof follows from martingale inequalities.

The advantage of concentration inequalities is that they hold for any fixed

N . However, they do not provide in general the optimal asymptotic speed. Such

optimal constants are provided by moderate deviations principle, or by large

deviations principle. Recall that a sequence of laws (PN , N ≥ 0) on a Polish

space Ξ satisfies a large deviation with good rate function I : Ξ → R+ and speed

aN going to infinity with N if and only if the level sets {x : I(x) ≤ M},M ≥ 0,

of I are compact and for all closed set F

lim supN→∞

a−1N log PN (F ) ≤ − inf

FI

whereas for all open set O

lim infN→∞

a−1N log PN (O) ≥ − inf

OI.

Large deviation results for the spectral measure of Wigner ensembles are

still only known for the Gaussian ensembles since their proof is based on the

explicit joint law of the eigenvalues P(β)N . This question was studied in [Ben97],

in relation with Voiculescu’s non-commutative entropy. The latter is defined as

the real-valued function on the set M1(R) of probability measures on the real

line given by

Σ(µ) =

{ ∫ ∫

log |x − y|dµ(x)dµ(y) if∫

log |x|dµ(x) < ∞ ,−∞ else .

(21.2.4)

Theorem 21.2.4 Let

Iβ(µ) =

{

β2

∫

x2dµ(x) − β2 Σ(µ) − cβ if

∫

x2dµ(x) < ∞,∞ else ,

with cβ = infν∈M1(R){β2

∫

x2dν(x) − β2 Σ(ν)}.

(21.2.5)

Then, the law of LN under P(β)N , as an element of M1(R) equipped with the

weak topology, satisfies a large deviation principle in the scale N 2, with good

rate function Iβ.

8 CHAPTER 21. WIGNER MATRICES

A moderate deviations principle for the spectral measure of the GUE or

GOE is also known, giving a sharp rate for the decrease of deviations from the

semi-circle law in a smaller scale.

Theorem 21.2.5 For any sequence aN → 0 so that NaN → ∞, the sequence

NaN (LN ([x,∞))−σ([x,∞))) in L1c(R) equipped with the Stieljes’-topology and

the corresponding cylinder σ-field satisfies the large deviation principle in the

scale (NaN )2 and with good rate function

J(F ) = sup{∫

h′(x)F (x)dx − 1

2

∫ 1

0(h′)2(

√sx)σ(dx)} .

Here the supremum is taken over the complex vector field generated by the Stielt-

jes functions f(x) = (z − x)−1, z ∈ C and the Stieltjes’-topology is the weak

topology with respect to derivatives of such functions.

This moderate deviation result does not have yet a fully universal version for

general Wigner ensembles. It has however been generalized to Gaussian divisi-

ble matrices (21.1.3) with a deterministic self-adjoint matrix VN with converging

spectral measure [Dem01] and to Bernouilli random matrices [Dor09].

21.2.4 The local semi-circle law and delocalization of the eigen-

vectors

A crucial result has been obtained much more recently, by Erdos, Schlein and

Yau [Erd08, Theorem 3.1] proving that the convergence to the semi-circle law

holds also locally, namely on more or less any scale larger than the typical

spacings between the normalized eigenvalues which is of order N−1. More

precisely, they consider the case where the distribution of the entries has sub-

exponential tails, and in the case of complex entries when the real and the

imaginary part is independent. They then showed the following theorem.

Theorem 21.2.6 For an interval I ⊂] − 2, 2[, let NI be the number of eigen-

values of XN/√

N which belong to I. Then, there exists positive constants c, C

so that for all κ ∈ (0, 2), all δ ≤ cκ, any η > (log N)4

N sufficiently small, we have

P

(

sup|E|≤2−κ

∣

∣

∣

∣

N[E−η,E+η]

2Nη− ρsc(E)

∣

∣

∣

∣

> δ

)

≤ Ce−cδ2√

Nη . (21.2.6)

Amazingly, this local convergence, concentration inequalities, independence

and equi-distribution of the entries, entails the delocalization of the eigenvectors

of XN , namely it was shown in [Erd08] that

21.2. GLOBAL PROPERTIES 9

Corollary 21.2.1 Under the same hypotheses, for any κ > 0 and K > 0, there

exists positive finite constants C, c such that for all N ≥ 2,

P

(

∃v so that XNv=√

Nµv, ‖v‖2 =1, µ∈ [−2 + κ, 2−κ] et ‖v‖∞≥ (log N)92

N12

)

is bounded above by Ce−c(log N)2 .

21.2.5 The limits of universality: the spectral measure of heavy

tailed Wigner random matrices

This brief section shows that there are natural obvious limits to the univer-

sality properties of Random Matrix Theory. When the second moment of the

distribution of the entries is infinite, the global behavior of the spectrum cannot

be expected to be similar to the Gaussian invariant cases, an indeed it changes

dramatically. Let us assume more precisely that the entries (Zi,j)i≤j , (Yi)i≥0 are

independent and have the same distribution P on R belonging to the domain

of attraction of an α-stable law. This means that there exists a slowly varying

function L so that

P (|x| ≥ u) =L(u)

uα

with α < 2. We then let

aN := inf{u : P (|x| ≥ u) ≤ 1

N}

which is of order N1α �

√N for α < 2. Note that aN is the order of magnitude

of the largest entries on a row (or column) of the matrix XN or of the sum of

the entries on a row (centered if α < 1). Then, it was stated in [Bou94] (see

Chapter 13 of this handbook) and proved in [Ben08] (see [Bel09] for the proof

of the almost sure convergence, and [Bord09] for another approach) that the

eigenvalues of XN are of order aN and that the spectral measure of XN correctly

renormalized converges towards a probability measure which is different from

the semi-circle law. More precisely we have

Theorem 21.2.7 Let α ∈ (0, 2) and put LN = N−1∑N

i=1 δa−1N λi

. Then

• LN converges weakly to a probability measure µα almost surely.

• µα is symmetric, has unbounded support, has a smooth density ρα which

satisfies

ρα(x) ∼ Lα

|x|α+1|x| → ∞.

10 CHAPTER 21. WIGNER MATRICES

21.3 Local properties in the bulk

The analysis of the local properties of the eigenvalues of XN in the bulk of

the spectrum goes back to Gaudin, Dyson and Mehta, among others, for the

Gaussian ensembles. The asymptotic behavior of the probability that no eigen-

values belong to an interval of width N−1, the asymptotic distribution of the

typical spacing between two nearest eigenvalues, the asymptotic behavior of the

k-points correlation functions are, for instance, well understood [Meh04]. The

generalization of these results to large classes of non-invariant Wigner ensem-

bles has been a major challenge for a long time. (as an anecdotal evidence of

this, it could be noted that during the 2006 conference in Courant Institute in

honor of Percy Deift’s 60th birthday, a panel of five experts all quoted this as

the main open question of Random Matrix Theory).

The first universality result in his direction was obtained by K.Johansson

[Joh01] for the correlation functions of complex Gaussian divisible entries (21.1.3).

Johansson’s proof follows and expands on an idea of Brezin and Hikami [Bre97].

This universality result has been recently extended to real entries by using dy-

namics and Dyson Brownian motion (21.1.4) by Erdos, Ramirez, Schlein, Yau

and Yin [Erd09a, Erd09b].

It was only very recently generalized to a general case of non-invariant

ensembles (with an assumption of sub-exponential moments and the same four

first moments as Gaussian variables) by Tao and Vu [Tao09a]. Combining these

two sets of results, one can prove the universality of the local statistics of the

eigenvalues in the bulk provided the entries have sub-exponential tails (and at

least 3 points in their support in the real case). We shall detail these results

in the next section, about spacing distributions and refer to the original papers

for correlation functions.

21.3.1 Spacings in the bulk for Gaussian Wigner ensembles

Let us first concentrates on a sample of local results for the Hermitian case.

Based on the fact that the law P(2)N is determinantal, the following asymptotics

are well known, see [Tra94], with ρsc the density of the semi-circle law.

Theorem 21.3.1 • For any x ∈ (−2, 2), any k ∈ N, the joint law of k (un-

ordered) rescaled eigenvalues ηi = N12 ρsc(x)−1(λi −N

12 x)1≤i≤k converges

vaguely towards the measure which is absolutely continuous with respect

to Lebesgue measure and with density

ρ(k)(η1, . . . , ηk) = Ck det ((S(ηi, ηj))1≤i,j≤k)

with S the Sine kernel

S(y1, y2) :=sin(y1 − y2)

π(y1 − y2).

21.3. LOCAL PROPERTIES IN THE BULK 11

• For any compact set B,

limN→∞

P(2)N

(

λi 6∈ N12 x + N− 1

2 ρsc(x)B, i = 1, . . . , N)

= ∆(B,S)(21.3.1)

with ∆ the Fredholm determinant

∆(B,S) := 1 +

∞∑

k=1

(−1)k

k!

∫

A· · ·∫

Adet(

(S(xi, xj))ki,j=1

)

k∏

i=1

dxi .

This result implies, see e.g. [Meh04, Appendix A.8] (or a more complete proof in

[And09, Theorem 4.2.49] together with [Dei99] which gives the uniform asymp-

totics of the Hermite kernel towards the Sine kernel on intervals with width

going to infinity with N) that the empirical spacing distribution between two

eigenvalues, near a given point in the bulk, converges. More precisely, we give

the statement for spacings near the origin. Define the Gaudin distribution by

PGaudin([2t,+∞)) = −C∂t∆(S,1(−t,t)c)

and consider a sequence lN increasing to infinity, such that l(N) = o(N).

Theorem 21.3.2 The number of eigenvalues of N− 12 XN at a distance less

than lN/N whose nearest neighbors spacing is smaller than N−1πs, divided by

lN , converges in probability towards PGaudin([0, s]).

These results generalize to the real case, even though the law P(1)N is not

determinantal anymore. For instance one has, see e.g. [And09, Section 3.1.2]

Theorem 21.3.3 There exists an increasing function F1 : R+ → R

+, F1(0) =

0, F1(∞) = 1 so that

1 − F1(t) = limN→∞

P(1)N

(

∩1≤i≤N{λNi ∈ 1√

N(− t

2,t

2)c})

Explicit formulae for F1 are known, see e.g. [And09, Theorem 3.1.6].

21.3.2 Universality for Gaussian divisible Wigner matrices

We will consider matrices given by (21.1.3) with VN a Wigner matrix. In the

complex case, we shall assume that the real and imaginary part of the entries of

VN are independent and that their common distribution have sub-exponential

tails. Then, we have

Theorem 21.3.4 For any ε ∈ (0, 1], the results of Theorem 21.3.1 extend to

the eigenvalues of XN =√

εGN +√

1 − εVN .

12 CHAPTER 21. WIGNER MATRICES

This result was proved by Johansson [Joh01] in the Hermitian case, based

on the fact that the density of the eigenvalues of XN is then explicitly given by

the Harich-Chandra-Itzykson-Zuber integral. It was initially valid only in the

middle of the bulk, i.e. very near the center of the semi-circle, but has been

later extended to the whole bulk. He showed more recently [Joh10] that the

existence of the second moment is sufficient to grant the result. In the real-

symmetric case, such a formula does not exist and the result was only recently

proved by Erdos, Schlein and Yau by noticing that the entries of XN can be

seen has the evolution of Brownian motions starting from the entries of VN

and taken at time − log ε, see (21.1.4). Based on this fact, Erdos, Schlein and

Yau developed techniques coming from hydrodynamics theory to show that a

small time evolution of the Brownian motion is sufficient to guarantee that the

correlation functions are close to equilibrium, given by the Gaussian matrix.

In fact, they could show that this time can even be chosen going to zero with

N . The universality result of Theorem 21.3.4 is weaker than those stated in

Chapter 6 of this handbook where convergence of the density of the joint law

of the eigenvalues is required; for the time being such an averaging is needed

to apply Erdos, Schlein and Yau techniques.

21.3.3 Universality and the four moments theorem

The approach proposed by Tao and Vu to prove universality follows Linden-

berg’s replacement argument, which is classical for sums of i.i.d random vari-

ables. The idea is to show that the eigenvalues of XN are smooth functions of

the entries so that one can replace one by one the entries of a Wigner matrix by

the entries of another Wigner matrix (for which we can control the local prop-

erties of the spectrum because they are Gaussian or Gaussian divisible) and

control the difference of the expectations by o(N−2) if the four first moments

are the same. The statement of the result is more precisely the following. Let us

consider two Wigner matrices XN and X ′N whose entries have sub-exponential

tails (in fact a sufficiently large number of moments should be enough) and, in

the complex case, with independent imaginary and real part. We assume that

the moments C(`, p) = E[<(Xij)`=(Xij)

p] are the same for XN and X ′N for all

` + p ≤ 4. We denote by λ1(MN ) ≤ λ2(MN ) · · · ≤ λN−1(MN ) ≤ λN (MN ) the

ordered eigenvalues of a Hermitian N × N matrix MN . Then, Theorem 15 of

[Tao09a] states that

Theorem 21.3.5 For any c0 > 0 sufficiently small, any ε ∈ (0, 1) and k ≥ 1,

any function F : Rk → R such that

|∇jF (x)| ≤ N c0

21.3. LOCAL PROPERTIES IN THE BULK 13

for all 0 ≤ j ≤ 5 and x ∈ Rk, we have for any i1, . . . , ik ∈ [εN, (1 − ε)N ],

∣

∣

∣E[

F (λij (√

NXN ), 1 ≤ j ≤ k)]

−E[

F (λij (√

NX ′N ), 1 ≤ j ≤ k)

]∣

∣

∣≤ N−c0 .

One can then take X ′N to be a Gaussian or a Gaussian divisible matrix to deduce

from the previous results on such matrices that any Wigner matrix having the

same moments of order smaller than four have asymptotically the same spacing

distributions in the bulk. It is not hard to see [Tao09a, Corollary 30] that

one can always match the first four moments with those of a Gaussian divisible

matrix provided the law of the entries have at least three points in their support.

In the Hermitian case, using a finer analysis based on the explicit formula for

Harich-Chandra-Itzykson-Zuber integral, one can prove [Erd09b] that Theorem

21.3.4 holds with ε of order N− 34 in which case one can always approximate the

first four moments of the entries with such a Gaussian divisible law to get rid

of the ”three points in the support” hypothesis [Erd09c].

21.3.4 The extreme gaps

We saw in the preceding question that the typical spacing between normalized

eigenvalues in the bulk is of order N−1, and that the distribution of a typical

spacing is known and universal. What about the size and distribution of the

smallest and of the largest spacings?

For the GUE ensemble, it has recently been proved ([Ben10] after initial

results in [Vin]) that the smallest spacing between normalized eigenvalues in

the bulk, say in the interval (−2 + ε, 2 − ε) for a positive ε, is of order N−4/3.

The distribution of this smallest spacings is known. Moreover the point process

of the smallest spacings is asymptotically Poissonian.

We first consider the smallest gaps, studying the point process

χN =

N−1∑

i=1

δ(N4/3(λi+1−λi),λi)1|λi|<2−ε,

for any arbitrarily small fixed ε > 0. Then, Theorem 1.4 of [Ben10] states that

Theorem 21.3.6 As N → ∞, the process χN converges to a Poisson point χ

process with intensity

E[χ(A × I)] =

(

1

48π2

∫

Au2du

)(∫

I(4 − x2)2dx

)

for any bounded Borel sets A ⊂ R+ and I ⊂ (−2 + ε, 2 − ε).

In fact this result obtained in [Ben10] is the same one would get if one

assumed (wrongly) that the spacings are independent. The correlations between

14 CHAPTER 21. WIGNER MATRICES

the spacings are not felt across the macroscopic distance between the smallest

gaps.The following corollary about the smallest gaps is an easy consequence of

the previous theorem. Introduce t(n)1 < · · · < t

(n)k the k smallest spacings in I,

i.e. of the form λi+1−λi, 1 ≤ i ≤ n−1, with λi ∈ I, I = [a, b], −2 < a < b < 2.

Let

τ(n)k =

(∫

I(4 − x2)2dx/(144π2)

)1/3

t(n)k .

Corollary 21.3.1 For any 0 ≤ x1 < y1 < · · · < xk < yk, with the above

notations

P(

x` < n4/3 τ(n)` < y`, 1 ≤ ` ≤ k

)

−→n→∞

(

e−x3k − e−y3

k

)

k−1∏

`=1

(

y3` − x3

`

)

.

In particular, the kth smallest normalized space N 4/3τNk converges in law to τk,

with distribution

P (τk ∈ dx) =3

(k − 1)!x3k−1e−x3

dx.

For the largest spacings, the situation is less understood. Nevertheless it is

proven that the largest gaps normalized by√

logNN converges to a constant.

The universality question related to the extreme gaps is still open. Can

these results be generalized to non-invariant ensembles? The real-symmetric

case is not understood even for the GOE.

The size√

log NN of the largest spacing is natural, it is the same as one would

guess from the tail of the Gaudin distribution by making the ansatz that the

spacings are independent. But it is also interestingly small. Indeed this size√log NN is also the size of the standard deviation of the position of a given

eigenvalue in the bulk [Gus]. Thus the maximal spacing is not bigger than a

typical fluctuation of one eigenvalue! This is of course linked to the fact that

two adjacent eigenvalues have perfectly correlated Gaussian fluctuations. Let

us mention here that Gustavsson’s result mentioned above has been proved to

be universal [Tao09a].

21.4 Local properties at the edge

It has long been observed that extreme eigenvalues of random matrices tend

to stick to the bulk, and hence converge to the boundary of the limiting spec-

tral measure. We shall describe this phenomena, study the fluctuations of the

extreme eigenvalues and their universality. For heavy tailed distribution of

the entries, and as mentioned above in the case of the spectral measure, this

21.4. LOCAL PROPERTIES AT THE EDGE 15

universality breaks down for the behavior of the extreme eigenvalues. The ex-

treme eigenvalues are then Poissonian and the associated eigenvectors are very

localized.

We will also mention an interesting universality question related to the

sensitivity of the extreme eigenvalues to the addition of a finite rank matrix.

21.4.1 Convergence of the extreme eigenvalues

The convergence of the extremal eigenvalues of a Wigner matrix towards the

boundary of the support of the semi-circle law goes back to Furedi and Komlos

[Fur81]. The following result has been proved by [Bai99]

Theorem 21.4.1 Assume that the fourth moment of the distribution of the

entries is finite. Then, λN := max1≤i≤N λNi and λN := min1≤i≤N λN

i converge

to 2 and −2 in probability.

The proof relies on fine estimates of the moments of the averaged spectral

measure, at powers going to infinity with N faster than logarithmically. It is

not hard to see that the convergence in probability can be improved into an

almost sure convergence. This result breaks down if the fourth moment is not

finite.

21.4.2 Fluctuations of the extreme eigenvalues and the Tracy-

Widom law

For the Gaussian ensembles, one can again rely on the explicit joint law of

the eigenvalues to study precisely the fluctutations of the extreme eigenvalues.

This is again simpler for the GUE, based on the determinantal structure of the

law P(2)N . After the work of Mehta [Meh04], a complete mathematical analysis

was given by the works of P. Forrester [For93] and C. Tracy and H. Widom

[Tra94, Tra00]. The main result goes as follows.

Theorem 21.4.2 For β = 1 or 2, for any t ∈ R, the largest eigenvalue λN of

XN is such that

Fβ(t) := limN→∞

P(β)N

(

N16 (λN − 2

√N) ≤ t

)

with Fβ the partition function for the Tracy-Widom law.

This result can be generalized to describe the joint convergence of the k-th

largest eigenvalues.

16 CHAPTER 21. WIGNER MATRICES

21.4.3 Universality

Based on very fine combinatorial arguments A. Soshnikov [Sos99] showed that

the moments of the spectral measure up to order N23 are the same than those

of the Gaussian ensembles. This allowed him to show that

Theorem 21.4.3 Assume that the entries (Zi,j)i≤j and (Yi)i≥0 are i.i.d with

distribution which is symmetric and with sub-Gausian tail. Then,the results of

Theorem 21.4.2 extend to XN .

By approximation arguments, it was shown in [R06] that it is sufficient to

have the first 36 moments finite, whereas new combinatorial arguments allowed

in [Kho09] to reduce the assumption to the twelve first moments finite. It is

conjectured, and proved in the Gaussian divisible case [Joh10], that the optimal

assumption is to have four moments finite. However, the hypothesis that the

law is symmetric could not be completely removed using this approach, see

[PS07] for an attempt in this direction.

21.4.4 Universality and the four moments theorem

T.Tao and V.Vu have generalized their four moments theorem to deal with the

eigenvalues at the edge [Tao09b, Theorem 1.13].

Theorem 21.4.4 Let XN and X ′N be two Wigner matrices with entries with

sub-exponential tails and moments which match up to order four. Assume in the

case of complex entries that the real and the imaginary parts are independent.

Then, there exists a small constant c0 > 0 so that for any function F : Rk → R

such that

|∇jF (x)| ≤ N c0

for all 0 ≤ j ≤ 5 and x ∈ Rk, we have for any i1, . . . , ik ∈ [1, N ],

∣

∣

∣E[

F (λij (√

NXN ), 1 ≤ j ≤ k)]

−E[

F (λij (√

NX ′N ), 1 ≤ j ≤ k)

]∣

∣

∣≤ N−c0 .

As a corollary, we have [Tao09b, Theorem 1.16].

Corollary 21.4.1 Under the assumptions of the previous theorem, let k be a

fixed integer number and XN a Wigner matrix with centered entries with sub-

exponential tails. Assume that they have the same covariance matrix as the

GUE (resp. GOE) and that all third moments vanish. Then the joint law of

((λN (XN/√

N) − 2)N23 , . . . , (λN−k(XN/

√N) − 2)N

23 ) converges weakly to the

same limit as for the GUE (resp. GOE).

Note that the advantage of this approach is that it does not require the sym-

metry of the distribution of the entries as Soshnikov’s result did.

21.4. LOCAL PROPERTIES AT THE EDGE 17

21.4.5 Extreme eigenvalues of heavy tailed Wigner matrices

Let us now consider what happens when the entries have no finite moments of

order four but are in the domain of attraction of an α-stable law as in section

21.2.5. Then, the behaviour of the extreme eigenvalues are dictated by the

largest entries of the matrix, and therefore the point process of the extreme

eigenvalues, once correctly normalized, converges to a point process, as in the

extreme value theory of independent variables. More precisely, assume that the

(Zi,j, i ≤ j) and (Yi, i ≥ 0) are i.i.d with law P and let

bN = inf{x : P (u : |u| ≥ x) ≤ 2

N(N + 1)}

which is of order N2α . Then it was shown in [Auf09], generalizing a result from

[Sos04], that

Theorem 21.4.5 Take α ∈ (0, 4) and for α ∈ [2, 4) assume that the entries

are centered. Then, the point process

PN =∑

i≥0

1λi≥0δb−1N λi

converges in distribution to the Poisson point process on (0,∞) with intensity

ρ(x) = αx1+α .

This result simply states that the largest eigenvalues behave as the largest

entries of the matrix.

21.4.6 Non universality and Extreme eigenvalues of finite rank

perturbation of Wigner matrices

In [Baik05], the authors consider the effect on the largest eigenvalue of adding

a finite rank matrix to a Gaussian sample covariance matrix. The phenomena

they observed, namely a phase transition in the asymptotic behavior and fluc-

tuations of the extreme eigenvalues, also happens when one considers Wigner

matrices, see e.g. [Cap09a]. A finite rank perturbation can pull some eigenval-

ues away from the bulk.

We will dwell only on the simplest case, where the rank of the perturbation

is one. But more is known, see e.g. [Fer07, Cap09a, Bai08]. Consider the

deformed Wigner matrix

MN =1√N

XN + AN

with XN a Wigner matrix with independent entries with symmetric law µ

satisfying Poincare inequality and AN a deterministic rank-one matrix. We

will look at the two extreme cases.

18 REFERENCES

First let AN = θN JN , where θ > 0 and JN is the matrix whose entries are

all ones, see e.g.[Fer07]. Obviously AN is rank-one and its eigenvalues are 0

and θ. The only non trivial eigenvector of AN is maximally delocalized. When

the parameter θ is small enough (θ < 1), the perturbation has no influence

on the top of the spectrum, the top eigenvalue λ1 of MN ”sticks to the bulk”,

i.e. λ1 converges to 2, the edge of the bulk. If XN is Gaussian it is proved

that the fluctuations are also unaffected by the perturbation, i.e that (λ1 −2)N2/3 converges to the Tracy-Widom distribution. This result is expected to

be universal. When the parameter θ is large enough (θ > 1), the top eigenvalue

λ1 is pulled away from the bulk, it converges to ρθ = θ+ 1θ > 2. Moreover in this

case the fluctuations of the top eigenvalue are in the scale√

N and Gaussian,

i.e√

N(λ1 − ρθ) converges to a Gaussian distribution. This result is universal.

But if one now chooses a rank-one perturbation with a very localized eigen-

vector, the situation is quite different. Let

AN = diag(θ

N, 0, . . . , 0)

Then again AN is rank-one and its eigenvalues are 0 and θ. The only non

trivial eigenvector of AN is maximally localized. In this case again when θ

is small enough (θ < 1), the perturbation has no influence on the top of the

spectrum, the top eigenvalue λ1 of MN ”sticks to the bulk”, i.e. λ1 converges

to 2, the edge of the bulk. Its fluctuations are Tracy-Widom in the invariant

case and expected to also be in the non-invariant cases. When the parameter

θ is large enough (θ > 1), the top eigenvalue λ1 is pulled away from the bulk,

it converges to ρθ > 2. But now the fluctuations are no longer Gaussian,√N(λ1−ρθ) converges in distribution to the convolution of a Gaussian law and

of µ. This limiting law thus remembers the law of the entries and this result is

definitely not universal. The reason is quite clear, and is due to the fact that,

because of the localization of the top eigenvector of the perturbation AN , the

top eigenvalue of MN remembers very much one entry of the matrix XN .

This interesting non-universal behavior has been studied in more general

cases in [Cap09b].

Acknowledgements: This work was supported by grant ANR-08-BLAN-

0311-01.

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